Energy-Driven Pattern Formation

Many physical systems can be modelled using a nonconvex bulk energy regularized by higher-order terms or surface energy. Such systems tend to form complex patterns of domains and walls.

It is natural to explore (a) the internal structures of walls, (b) the overall patterns of domains, and (c) the nucleation of new domains and walls due to thermal fluctuation. When a pattern is driven by energy minimization we can "explain" it variationally, by showing that no other pattern achieves a lower energy. I will discuss several examples of this type, including (i) a bound on the rate of energy-driven coarsening (work with Otto); (ii) the internal structure of a cross-tie wall (work by Alouges, Riviere, and Serfaty), and (iii) thermally-activated switching modelled by action minimization (work with Otto, Reznikoff, and Vanden-Eijnden).